What is the New Definition of Conformal Transformations?

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SUMMARY

Conformal transformations are now defined as smooth mappings \(\phi:U\rightarrow V\) that satisfy the condition \(\phi^{*}g'=\Omega^{2}g\), where \(\Omega:U\rightarrow R_{+}\) is a smooth function. This new definition emphasizes the importance of the tangent map \(T\phi\) in relating the geometries of the two spaces involved. The pullback operation is essential for comparing geometric objects in different spaces, ensuring that the transformation is well-defined across regions.

PREREQUISITES
  • Understanding of differential geometry concepts, particularly metrics and smooth mappings.
  • Familiarity with the notation and operations involving pullbacks in differential geometry.
  • Knowledge of tangent spaces and their role in transformations.
  • Basic grasp of positive functions and their implications in geometric contexts.
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  • Study the properties of smooth functions and their applications in differential geometry.
  • Explore the concept of pullbacks in more detail, particularly in the context of metric tensors.
  • Investigate the implications of tangent maps in conformal geometry.
  • Learn about the applications of conformal transformations in physics, especially in general relativity.
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Mathematicians, physicists, and students of differential geometry who are interested in advanced concepts of transformations and their applications in various fields.

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Conformal transformations as far as I knew are defined as [itex]g_{mn}\rightarrow g'_{mn}=\Omega g_{mn}[/itex].

Now I come across a new definition, such that a smooth mapping [itex]\phi:U\rightarrow V[/itex] is called a conformal transformation if there exist a smooth function [itex]\Omega:U\rightarrow R_{+}[/itex] such that [itex]\phi^{*}g'=\Omega^{2}g[/itex] where [itex]\phi^{*}g'(X,Y):=g'(T\phi(X),T\phi(Y))[/itex] and [itex]T\phi :TU\rightarrow TV[/itex] denotes the tangent map of [itex]\phi[/itex].

I can't really make sense of this. Why do we need the derivative of the map to define the transformation?
 
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You can only compare objects that exist on the same space (or region U in this case). So the pullback is needed to accomplish that.
 

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